Determining If a Number is Prime

Date: 08/12/2004 at 21:24:52
From: Daryl
Subject: Prime Numbers


Is there a formula for finding if a number is prime?  I've written a
program that will tell you if a number is prime, but it is slow for
larger numbers because it checks all numbers between 1 and the number
to see if they are factors of the larger number.  As you can see,
checking a number in the billions would result in billions of
divisions, which takes time.

I've tried looking online, but haven't been able to find anything.

Date: 08/13/2004 at 12:21:49
From: Doctor Vogler
Subject: Re: Prime Numbers

Hi Daryl,

Thanks for writing to Dr Math.  There are several "formulas."  They
are called "primality tests."  First there are the "pseudoprimality
tests" which can quickly tell you if a number is NOT prime, but they
can not guarantee that a number IS prime.  The Fermat test is the
best-known of these.  The true primality tests are more complicated
(which is why you usually start by checking if the number is clearly
composite first, with a pseudoprimality test) but they can tell you
conclusively whether a number is prime.  In fact, a recent discovery
was a "polynomial-time deterministic primality test" which means that
it is "fast" by a particular technical definition of "fast."  Other
non-deterministic algorithms are more effective or more efficient, but
this one is interesting theory.  For precise descriptions of 
particular primality tests, search Google for "primality test."  See
also the account on MathWorld at

  Primality Test
    http://mathworld.wolfram.com/PrimalityTest.html 

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/